Compound Poisson Cascades

Multiplicative processes and multifractals proved useful in various applications ranging from hydrodynamic turbulence to computer network traffic. Placing multifractal analysis in the more general framework of infinitely divisible laws, we design processes which possess at the same time stationary increments as well as multifractal and more general infinitely divisible scaling over a continuous range of scales. The construction is based on a work by Barral & Mandelbrot [4] where a Poissonian geometry was introduced to allow for continuous multiplication. As they possess compound Poissonian statistics we term the resulting processes Compound Poisson Cascades. We explain how to tune their correlation structure, as well as their scaling properties, and hint at how to go beyond pure power law scaling behaviours towards more general infinitely divisible scaling. Further, we point out that these cascades represent but the most simple and most intuitive case out of an entire array of infinitely divisible cascades allowing to construct general infinitely divisible processes with interesting scaling properties. Modelling complexity through scaling. Scale invariance and related phenomena have received considerable attention in the past from the point 1P. Chainais acknowledges the support of Rice university during his stay in Houston in November 2001 during which this work was further developed. 2Support for R. Riedi comes in part from DARPA F30602-00-2-0557, NSF ANI-00099148 and from Texas Instruments. R. Riedi gratefully acknowledges the support of CNRS and ENS Lyon during his visit to ENS Lyon in May 2001 during which much of this work was developed. 3This work has been partly supported by grant ACI Jeune Chercheur 2329, 1999, from the French MENRT.

[1]  R. Riedi,et al.  Multifractal products of stochastic processes: construction and some basic properties , 2002, Advances in Applied Probability.

[2]  F. Schmitt,et al.  Stochastic equations generating continuous multiplicative cascades , 2001, cond-mat/0102346.

[3]  B. Mandelbrot,et al.  Multifractal products of cylindrical pulses , 2002 .

[4]  Walter Willinger,et al.  Self-Similar Network Traffic and Performance Evaluation , 2000 .

[5]  P. Abry,et al.  SCALE INVARIANT INFINITELY DIVISIBLE CASCADES , 2004 .

[6]  Patrice Abry,et al.  Intermittency and coherent structures in a swirling flow: A wavelet analysis of joint pressure and velocity measurements , 1999 .

[7]  Editors , 1986, Brain Research Bulletin.

[8]  Y. Gagne,et al.  Velocity probability density functions of high Reynolds number turbulence , 1990 .

[9]  E. Bacry,et al.  Multifractal random walk. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  Rudolf H. Riedi,et al.  Wavelet analysis of fractional Brownian motion in multifractal time , 1999 .

[11]  Emmanuel Bacry,et al.  Random cascades on wavelet dyadic trees , 1998 .

[12]  D. Veitch,et al.  Infinitely divisible cascade analysis of network traffic data , 2000, 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.00CH37100).

[13]  Walter Willinger,et al.  On the self-similar nature of Ethernet traffic , 1993, SIGCOMM '93.

[14]  J. Kahane,et al.  Sur certaines martingales de Benoit Mandelbrot , 1976 .

[15]  Pierre Chainais.  Cascades log-infiniment divisibles et anlayse multirésolution : application à l'étude des intermittences en turbulences , 2001 .

[16]  Alain Arneodo,et al.  Experimental analysis of self-similarity and random cascade processes : Application to fully developed turbulence data , 1997 .

[17]  Metin Akay,et al.  Nonlinear Biomedical Signal Processing, Dynamic Analysis and Modeling , 2000 .

[18]  Benoit B. Mandelbrot,et al.  Fractals and Scaling in Finance , 1997 .